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The F-rational signature and drops in the Hilbert–Kunz multiplicity

Melvin Hochster and Yongwei Yao

Vol. 16 (2022), No. 8, 1777–1809

Let (R,𝔪) be a Noetherian local ring of prime characteristic p. We define the F-rational signature of R, denoted by r (R), as the infimum, taken over pairs of ideals I J such that I is generated by a system of parameters and J is a strictly larger ideal, of the drops e HK (I,R)) e HK (J,R) in the Hilbert–Kunz multiplicity. If R is excellent, then R is F-rational if and only if r (R) > 0. The proof of this fact depends on the following result in the sequel: Given an 𝔪-primary ideal I in R, there exists a positive δI + such that, for any ideal J I, e HK (I,R) e HK (J,R) is either 0 or at least δI. We study how the F-rational signature behaves under deformation, flat local ring extension, and localization.

F-rational signature, F-signature, Hilbert–Kunz multiplicity, Frobenius, Cohen–Macaulay, Gorenstein, regular
Mathematical Subject Classification 2010
Primary: 13A35
Secondary: 13C13, 13H10
Received: 19 September 2017
Revised: 8 August 2021
Accepted: 22 November 2021
Published: 29 November 2022
Melvin Hochster
Department of Mathematics
University of Michigan
Ann Arbor
United States
Yongwei Yao
Department of Mathemaics and Statistics
Georgia State University
United States